Anderson-Smith studied weakly prime ideals for a commutative ring with identity. Hirano, Poon and Tsutsui studied the structure of a ring in which every ideal is weakly prime for rings, not necessarily commutative. In this note we give some more properties of weakly prime ideals in noncommutative rings. We introduce the notion of a weakly prime radical of an ideal. We initiate the study of weakly completely prime ideals and investigate rings for which every proper ideal is weakly completely prime.
{"title":"WEAKLY PRIME AND WEAKLY COMPLETELY PRIME IDEALS OF NONCOMMUTATIVE RINGS","authors":"N. Groenewald","doi":"10.24330/ieja.768127","DOIUrl":"https://doi.org/10.24330/ieja.768127","url":null,"abstract":"Anderson-Smith studied weakly prime ideals for a commutative ring with identity. Hirano, Poon and Tsutsui studied the structure of a ring in which every ideal is weakly prime for rings, not necessarily commutative. In this note we give some more properties of weakly prime ideals in noncommutative rings. We introduce the notion of a weakly prime radical of an ideal. We initiate the study of weakly completely prime ideals and investigate rings for which every proper ideal is weakly completely prime.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49419172","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We prove that there is no perfect binary polynomial R that is the sum of two appropriate powers, besides, possibly R = P +1 with P irreducible. The proofs follow from analogue results involving the ABC-theorem for polynomials and a classical identity.
． 我们证明了不存在两个适当的幂和的完美二元多项式R，并且可能R = P +1且P不可约。这些证明来自多项式的abc定理和一个经典恒等式的类似结果。
{"title":"MASON-STOTHERS THEOREM AND PERFECT BINARY POLYNOMIALS","authors":"L. Gallardo","doi":"10.24330/ieja.768086","DOIUrl":"https://doi.org/10.24330/ieja.768086","url":null,"abstract":". We prove that there is no perfect binary polynomial R that is the sum of two appropriate powers, besides, possibly R = P +1 with P irreducible. The proofs follow from analogue results involving the ABC-theorem for polynomials and a classical identity.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43905288","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, graded rings are $S$-graded rings inducing $S,$ that is, rings whose additive groups can be written as a direct sum of a family of their additive subgroups indexed by a nonempty set $S,$ and such that the product of two homogeneous elements is again a homogeneous element. As a generalization of the recently introduced notion of a $UJ$-ring, we define a graded $UJ$-ring. Graded nil clean rings which are graded $UJ$ are described. We also investigate the graded $UJ$-property under some graded ring constructions.
{"title":"ON GRADED UJ-RINGS","authors":"E. Ilić-Georgijević","doi":"10.24330/ieja.768259","DOIUrl":"https://doi.org/10.24330/ieja.768259","url":null,"abstract":"In this paper, graded rings are $S$-graded rings inducing $S,$ that is, rings whose additive groups can be written as a direct sum of a family of their additive subgroups indexed by a nonempty set $S,$ and such that the product of two homogeneous elements is again a homogeneous element. As a generalization of the recently introduced notion of a $UJ$-ring, we define a graded $UJ$-ring. Graded nil clean rings which are graded $UJ$ are described. We also investigate the graded $UJ$-property under some graded ring constructions.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46701974","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we present some commutativity theorems for prime rings R with involution ∗ of the second kind in which endomorphisms satisfy certain algebraic identities. Furthermore, we provide examples to show that various restrictions imposed in the hypotheses of our theorems are not superfluous. Mathematics Subject Classification (2020): 16N60, 16W10, 16W25
{"title":"ENDOMORPHISMS WITH CENTRAL VALUES ON PRIME RINGS WITH INVOLUTION","authors":"L. Oukhtite, H. E. Mir, B. Nejjar","doi":"10.24330/ieja.768202","DOIUrl":"https://doi.org/10.24330/ieja.768202","url":null,"abstract":"In this paper we present some commutativity theorems for prime rings R with involution ∗ of the second kind in which endomorphisms satisfy certain algebraic identities. Furthermore, we provide examples to show that various restrictions imposed in the hypotheses of our theorems are not superfluous. Mathematics Subject Classification (2020): 16N60, 16W10, 16W25","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44117791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we give a criterion of the Gorenstein property of the Ehrhart ring of the stable set polytope of an h-perfect graph: the Ehrhart ring of the stable set polytope of an h-perfect graph $G$ is Gorenstein if and only if (1) sizes of maximal cliques are constant (say $n$) and (2) (a) $n=1$, (b) $n=2$ and there is no odd cycle without chord and length at least 7 or (c) $ngeq 3$ and there is no odd cycle without chord and length at least 5.
{"title":"ON THE GORENSTEIN PROPERTY OF THE EHRHART RING OF THE STABLE SET POLYTOPE OF AN H-PERFECT GRAPH","authors":"Mitsuhiro Miyazaki","doi":"10.24330/IEJA.969935","DOIUrl":"https://doi.org/10.24330/IEJA.969935","url":null,"abstract":"In this paper, we give a criterion of the Gorenstein property of the Ehrhart ring of the stable set polytope of an h-perfect graph: the Ehrhart ring of the stable set polytope of an h-perfect graph $G$ is Gorenstein if and only if (1) sizes of maximal cliques are constant (say $n$) and (2) (a) $n=1$, (b) $n=2$ and there is no odd cycle without chord and length at least 7 or (c) $ngeq 3$ and there is no odd cycle without chord and length at least 5.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45223580","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Motivated by q-shuffle products determined by Singer from q-analogues of multiple zeta values, we build in this article a generalisation of the shuffle and stuffle products in terms of weak shuffle and stuffle products. Then, we characterise weak shuffle products and give as examples the case of an alphabet of cardinality two or three. We focus on a comparison between algebraic structures respected in the classical case and in the weak case. As in the classical case, each weak shuffle product can be equipped with a dendriform structure. However, they have another behaviour towards the quadri-algebra and the Hopf algebra structure. We give some relations satisfied by weak stuffle products.
{"title":"Weak stuffle algebras","authors":"Cécile Mammez","doi":"10.24330/ieja.1060709","DOIUrl":"https://doi.org/10.24330/ieja.1060709","url":null,"abstract":"Motivated by q-shuffle products determined by Singer from q-analogues of multiple zeta values, we build in this article a generalisation of the shuffle and stuffle products in terms of weak shuffle and stuffle products. Then, we characterise weak shuffle products and give as examples the case of an alphabet of cardinality two or three. We focus on a comparison between algebraic structures respected in the classical case and in the weak case. As in the classical case, each weak shuffle product can be equipped with a dendriform structure. However, they have another behaviour towards the quadri-algebra and the Hopf algebra structure. We give some relations satisfied by weak stuffle products.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45667611","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $M$ be a module over a commutative ring $R$. The annihilating-submodule graph of $M$, denoted by $AG(M)$, is a simple graph in which a non-zero submodule $N$ of $M$ is a vertex if and only if there exists a non-zero proper submodule $K$ of $M$ such that $NK=(0)$, where $NK$, the product of $N$ and $K$, is denoted by $(N:M)(K:M)M$ and two distinct vertices $N$ and $K$ are adjacent if and only if $NK=(0)$. This graph is a submodule version of the annihilating-ideal graph and under some conditions, is isomorphic with an induced subgraph of the Zariski topology-graph $G(tau_T)$ which was introduced in (The Zariski topology-graph of modules over commutative rings, Comm. Algebra., 42 (2014), 3283--3296). In this paper, we study the domination number of $AG(M)$ and some connections between the graph-theoretic properties of $AG(M)$ and algebraic properties of module $M$.
{"title":"DOMINATION NUMBER IN THE ANNIHILATING-SUBMODULE GRAPH OF MODULES OVER COMMUTATIVE RINGS","authors":"H. Ansari-Toroghy, S. Habibi","doi":"10.24330/IEJA.969902","DOIUrl":"https://doi.org/10.24330/IEJA.969902","url":null,"abstract":"Let $M$ be a module over a commutative ring $R$. The annihilating-submodule graph of $M$, denoted by $AG(M)$, is a simple graph in which a non-zero submodule $N$ of $M$ is a vertex if and only if there exists a non-zero proper submodule $K$ of $M$ such that $NK=(0)$, where $NK$, the product of $N$ and $K$, is denoted by $(N:M)(K:M)M$ and two distinct vertices $N$ and $K$ are adjacent if and only if $NK=(0)$. This graph is a submodule version of the annihilating-ideal graph and under some conditions, is isomorphic with an induced subgraph of the Zariski topology-graph $G(tau_T)$ which was introduced in (The Zariski topology-graph of modules over commutative rings, Comm. Algebra., 42 (2014), 3283--3296). In this paper, we study the domination number of $AG(M)$ and some connections between the graph-theoretic properties of $AG(M)$ and algebraic properties of module $M$.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44749435","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the article, we perform a classification of algebras with dimensions $leq$ 3 and with the property that each element is colinear with its square. The classification is complete up to properties of the ground field.
{"title":"CLASSIFICATION OF THREE-DIMENSIONAL ZEROPOTENT ALGEBRAS","authors":"A. Cedilnik, Marjan Jerman","doi":"10.24330/ieja.662996","DOIUrl":"https://doi.org/10.24330/ieja.662996","url":null,"abstract":"In the article, we perform a classification of algebras with dimensions $leq$ 3 and with the property that each element is colinear with its square. The classification is complete up to properties of the ground field.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2020-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48580745","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
T. Breuer, L'aszl'o H'ethelyi, Erzs'ebet Horv'ath, B. Kulshammer
In Part I of this paper, we introduced a class of certain algebras of finite dimension over a field. All these algebras are split, symmetric and local. Here we continue to investigate their Loewy structure. We show that in many cases their Loewy length is equal to an upper bound established in Part I, but we also construct examples where we have a strict inequality.
{"title":"THE LOEWY STRUCTURE OF CERTAIN FIXPOINT ALGEBRAS, PART II","authors":"T. Breuer, L'aszl'o H'ethelyi, Erzs'ebet Horv'ath, B. Kulshammer","doi":"10.24330/ieja.969577","DOIUrl":"https://doi.org/10.24330/ieja.969577","url":null,"abstract":"In Part I of this paper, we introduced a class of certain algebras of finite dimension over a field. All these algebras are split, symmetric and local. Here we continue to investigate their Loewy structure. We show that in many cases their Loewy length is equal to an upper bound established in Part I, but we also construct examples where we have a strict inequality.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42378750","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We examine the Loewy series associated to the lattice of subalgebras of an extension of commutative rings.
研究了交换环扩展子代数格的Loewy级数。
{"title":"THE LOEWY SERIES OF AN FCP (DISTRIBUTIVE) RING EXTENSION","authors":"Gabriel Picavet, M. Picavet-L'Hermitte","doi":"10.24330/ieja.851985","DOIUrl":"https://doi.org/10.24330/ieja.851985","url":null,"abstract":"We examine the Loewy series associated to the lattice of subalgebras of an extension of commutative rings.","PeriodicalId":43749,"journal":{"name":"International Electronic Journal of Algebra","volume":" ","pages":""},"PeriodicalIF":0.6,"publicationDate":"2019-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45788031","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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